The prototype of statements on hyperbolicity of moduli spaces is that periods of curves map into bounded domains. More recent and more algebro-geometric examples are Shafarevich's conjecture on families of curves or that the moduli space of curves or of K3 surfaces are of (log-)general type, with possibly finitely many exceptions. On the differential geometry side, several results claim negativity of different metrics on the moduli space of Kähler-Einstein manifolds. The common theme about all such statements is that they show that some properties of curves of general type hold also for moduli spaces of varieties. These properties are referred to as hyperbolicity properties. The subject of my talk is the hyperbolicity of the moduli space of canonically polarized manifolds and of its compactification, the moduli space of stable schemes. I will talk about the available conjectures, results and my contributions to the topic. Notice also, that these are amongst the very few statements about the global geometry of general components of these moduli spaces.